A Mathematical Theory of Communication

25 Channel capacity with an average power limitation

1. State the capacity of a band-limited channel with white Gaussian noise (Theorem 17).
   \(C = W \log \frac{P + N}{N}\) for bandwidth \(W\), average signal power \(P\) and white noise power \(N\). To approach it, the transmitted signals must themselves resemble white noise.
   • Shannon, 1948, A Mathematical Theory of Communication (Bell System Technical Journal), §25
2. Describe Shannon’s ideal scheme for the white-noise channel.
   Take \(M = 2^s\) samples of white noise of duration \(T\) as codewords; send the one numbered by each group of \(s\) message bits; the receiver picks the codeword with the least RMS distance to what it received (the most probable signal). As \(T\) grows, the rate approaches \(W \log \frac{P+N}{N}\).
   • Shannon, 1948, A Mathematical Theory of Communication (Bell System Technical Journal), §25